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Simulating Gravity: Dark Matter and Gravitational Lensing in the Classroom
Jes Ford, Jared Stang, and Catherine Anderson
Citation: The Physics Teacher 53, 557 (2015); doi: 10.1119/1.4935771
View online: http://dx.doi.org/10.1119/1.4935771
View Table of Contents: http://scitation.aip.org/content/aapt/journal/tpt/53/9?ver=pdfcov
Published by the American Association of Physics Teachers
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Simulating Gravity: Dark Matter and
Gravitational Lensing in the Classroom
Jes Ford, Jared Stang, and Catherine Anderson, University of British Columbia & Science World, Vancouver, Canada
D
ark matter makes up most of the matter in the uniSolving for the orbital speed of the star yields
verse but very little of a standard introductory physics curriculum. Here we present our construction
(2)
and use of a spandex sheet-style gravity simulator to qualitatively demonstrate two aspects of modern physics related to
The orbital speed of the star increases with the amount of
dark matter. First, we describe an activity in which students
galactic mass M within the orbit and decreases with increasexplore the dependence of orbital velocities on the central
ing distance r. This agrees with intuition: More galactic mass
mass of a system, in a demonstration of how scientists
first discovered dark matter. Second, we discuss the
use of the gravity simulator
as a visualization of gravitational lensing, a current
astronomical technique for
mapping dark matter in
the sky. After providing the Fig. 1. The rotational velocity for stars in the Andromeda galaxy, M31, as a function of distance from
necessary background for
the center of the galaxy overlaid on an image of the galaxy.2 Outside of the luminous center, the rotation
speed curve is flat and does not decrease as 1/√r as would be expected from observations of
the phenomena of interest,
the luminous matter. (Image by Vera Rubin and Janice Dunlap, courtesy of Carnegie Institution DTM.)
we describe our construction of the gravity simulator
and detail our facilitation of these two activities. Together,
would be able to better hold (faster) stars in orbit, while stars
these activities provide a conceptual visualization of gravitafurther from the center will feel a smaller gravitational force
tional phenomena related to indirect detection techniques for
so that to be gravitationally bound they would need a smaller
studying dark matter.
orbital speed.
If all the mass in galaxies came from luminous matter
Background
(matter that interacts with electromagnetic waves, or light),
Dark matter and galaxy rotation curves
then we would expect the rotation speed of the outer stars to
Most of the matter in the universe is invisible dark matter,
decrease with
In conflict with this expectation, astronwhich can only be detected through its gravitational influomers observe that, outside the luminous center, the rotation
ence. Astronomers know dark matter must be present in galspeed of stars is approximately constant as a function of their
axies because the galaxies rotate much faster than would be
distance from the center of the galaxy (Fig. 1). From Eq. (2),
possible if their mass was composed only of things we can see,
in order for the rotation speed to be constant with r, the total
like stars and dust.
galactic mass M would have to increase linearly with r. Rubin,
The expected speed of rotation for stars in galaxies can
Ford, and Thonnard write: “The conclusion is inescapable
easily be derived using Newtonian physics. Consider a star
that non-luminous matter exists beyond the optical galaxy.”3
far from the center of a galaxy that orbits the center in a cirThis is dark matter.
cular orbit. The force holding the star in orbit is the attractive
gravitational force from all the mass inside the star’s orbit.
Gravitational lensing and the Einstein radius
Newton’s second law gives
According to Einstein’s theory of general relativity, dark
matter
exerts a gravitational influence on light particles (pho(1)
tons). This is a manifestation of gravitational lensing, wherein
light rays are deflected near massive objects. Anything maswhere m is the mass of the orbiting star, ac = v2/r is the censive can work—a star, a black hole, a galaxy, or a cluster of
tripetal acceleration of the star, G is the gravitational congalaxies all produce lensing. The result is that the images of
stant, M is the galactic mass inside the orbit of the star, and r
objects behind these lenses are altered and distorted, and
is the distance from the center of the galaxy to the star’s orbit.
astronomers can measure the lensing to infer the mass of the
gravitational lens.
DOI: 10.1119/1.4935771
THE PHYSICS TEACHER ◆ Vol. 53, DECEMBER 2015
557
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The gravity simulator
Fig. 2. Generic gravitational lensing geometry (left) compared
with the gravity simulator set up for measuring the Einstein
radius of the lens (right, described in paper below). The source
(represented by the black star) is directly behind the lens (black
circle) when β = 0. In this case, the angle at which the source can
be viewed is called the Einstein radius θE. For a symmetric lens,
the source will be lensed into an Einstein ring.
Fig. 3. Example of a nearly full Einstein Ring. (Image courtesy of
ESA/Hubble & NASA.)
A lensed background light source at angular position β
(relative to the lens) will appear instead to be located at angular position θ. The lens geometry is shown in the left diagram
of Fig. 2, yielding the lens equation θ = β + α, where α is the
deflection angle, given by4
(3)
Here c is the speed of light, we have assumed a point-like lens
of mass M, and the other geometrical quantities are defined
in Fig. 2. When a background light source is directly behind
a symmetric gravitational lens, its light can be projected into
an apparent circle around the lens—a phenomenon known as
an Einstein ring (Fig. 3). For this scenario we set β = 0 in the
lensing equation and, using b = Dl θ, derive the angular radius
of the Einstein ring to be
(4)
A more massive lens (larger M) will deflect light rays
more, leading to a larger Einstein radius θE.5 Astronomers use
gravitational lensing to map the dark matter in the sky.
558
The gravity simulator, as conceptualized here, consists of
a spandex sheet stretched over a circular frame, similar to a
trampoline. Upon placing masses on the fabric, gravitational
effects are visualized as a deformation of the fabric. The
trajectories of “test particles” (typically small marbles) are
curved due to the presence of masses, providing a representation of gravitational effects.
Several papers have investigated the motion of marbles on
these spandex sheet-style gravity simulators, finding that the
motion does not precisely (i.e., mathematically) reproduce
the motion of bodies moving under the influence of gravity.6,7
For example, for a spandex sheet that was under no tension
before the central mass was put into place, the usual Kepler
relation T2 ! r3 becomes instead T 3 ! r 2. 6 However, this
demonstration offers a visual and hands-on qualitative representation of gravitational forces which shares some essential
features with physical gravitational systems, thus making it a
valuable teaching resource for certain conceptual ideas (such
as those discussed in this paper).
Building the gravity simulator
In this section, we describe our construction of a gravity
simulator. Descriptions of alternative constructions are available.8,9
Our gravity simulator consists of a circular wooden frame
over which nylon LYCRA® is stretched. The materials we used
to construct the simulator are:
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etc.)
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We used a skill saw to cut the plywood into sections that
would form the outside circular edge, before sanding the sections until they were smooth. The wooden frame was glued
and screwed together, using wooden braces to join the sections of the circle.105IF17$QJQFTBOEKPJOUTXFSFVTFEUP
create removable legs for the apparatus. Once the table-like
structure was ready (Fig. 4), the nylon LYCRA sheet could
be overlaid and held in place by the spring clamps spaced
at equal intervals around the border (Fig. 5). It is important
that the LYCRA sheet stretches equally in both directions so
that the effect of a mass on the sheet is isotropic. The foam
pipe insulation can be secured around the outside edge of the
gravity simulator to help keep marbles on the simulator and
to make the edges safer for participants.
THE PHYSICS TEACHER ◆ Vol. 53, DECEMBER 2015
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students might compare the orbital speeds of marbles with
two different masses in the center of the simulator or they
could observe how the orbital speed changes as they slowly
increase the mass in the center.
Students should observe that even if the mass in the center
is not directly measurable (as is the case for dark matter), its
presence can be inferred and quantified by measuring orbital
velocities.
Fig. 4. The circular wooden frame of the gravity simulator. The
simulator stands 3 ft tall and the inner diameter is about 5 ft.
Fig. 5. The fully assembled gravity simulator.
Activities
Orbits and galaxy rotation curves
In this activity, students observe that the rotation speed
of orbiting marbles is larger for larger central masses.11 This
connects with the observations of galactic rotation curves:
Since the stars are moving faster than we expect based on
observations of luminous matter, astronomers conclude that
there is more (non-luminous) matter present in galaxies.
We used an inquiry-based approach, working in small
groups (eight) with students. We began by asking students
what they knew about gravity, exploring their prior knowledge before making any connections to the gravity simulator.12 Through facilitated discussion, we encouraged them
to ask questions that could be addressed with the gravity
simulator. Students are curious but not necessarily good at
developing answerable questions; this stage requires some
guidance to keep them on topic and within the constraints of
the device. We led them to the question of what affects the orbital motion of celestial bodies, and within that to the specific
experiment of deciding how orbital speeds depend on the
mass in the center of the gravity simulator. Depending on the
group, this experiment could look different. For example, the
Measuring the Einstein radius
In this activity, students develop intuition about gravitational lensing and qualitatively explore the relationship between the mass of a lens and its Einstein radius.
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tube anchored at a fixed angle to the horizontal—which
gives the “photons” (represented by marbles) a constant initial speed as they enter the simulator.13 A “telescope” (small
target container to catch the marbles) is positioned on the
opposite side of the gravity simulator, and a lensing mass is
placed in the center, as in the right panel of Fig. 2.14 Students
adjust the angle of the photon source until they can hit the
telescope, in analogy to actually observing the light from a
galaxy behind the lens. Since the lens is approximately halfway between the source and the telescope, the angle of the
photon tube is approximately equal to the angle of the photon
reaching the telescope. This is the Einstein radius θE.
This activity provides a concrete visualization of the path
light takes around a massive lens. Students should be able to
hit the telescope at the same angle on both sides of the lens,
and they should find that larger mass lenses will lead to larger
Einstein rings.
Further possibilities
When using the gravity simulator, we emphasized qualitative observations and encouraged students to create their own
experiments. In addition to the activities described above,
they enthusiastically came up with many “what if?” scenarios
to test. One of the most interesting of these was one student’s
idea for many of them to simultaneously toss entire handfuls
of marbles in different directions around the gravity simulator. This experiment simulates the formation of a solar system
or galaxy; as opposing marbles collide and fall to the center,
the system is eventually reduced to several marbles (or planets) all orbiting in the same direction.15
Making connections
To further explore lensing, we supplemented the above
gravity simulator activity with a gravitational lensing activity
using the glass bases of wine glasses to simulate gravitational
lensing effects.16,17 Light is bent (or refracted) by the glass,
in a similar way to light being bent by gravity near a very
massive lens.17 Experimenting with both gravitational lensing demonstrations in the same session allowed students to
visualize the effects of dark matter and gravitational lensing
with complementary lower-dimensional representations.
THE PHYSICS TEACHER ◆ Vol. 53, DECEMBER 2015
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(The gravity simulator removes one of the spatial dimensions
in the plane perpendicular to the observer-lens-source direction, while the glass lens demonstration essentially removes
the line-of-sight dimension.)
Several other teaching resources are available that focus
specifically on dark matter18 and gravitational lensing,17 and
there are many excellent web resources with additional details
on the fascinating mystery of dark matter.19
The activities described above require an initial investment to build the gravity simulator; once constructed, the device can be used for many years. Although we have described
two concrete activities aimed at the introductory physics
level, the gravity simulator lends itself to a plethora of different demonstration possibilities and can be used for groups at
many different levels20 to vividly illustrate various phenomena related to gravity or general relativity.
12.
13.
14.
15.
References
1.
5.
6.
7.
8.
9.
10.
11.
560
We assume a spherical orbit for the star and a spherical distribution of mass within the galaxy.
7FSB3VCJOi4FFJOHEBSLNBUUFSJOUIF"OESPNFEBHBMBYZw
Phys. Today 59 (12), 8 (2006).
7$3VCJO8,'PSE+SBOE/5IPOOBSEi3PUBUJPOBMQSPQerties of 21 Sc galaxies with a large range of luminosities and
radii, from NGC 4605 (R=4 kpc) to UGC 2885 (R=122 kpc),”
Astrophys. J. 238, 471–487 (1980).
+BNFT#)BSUMFGravity: An Introduction to Einstein’s General
Relativity (Addison-Wesley, 2003).
In the universe, changing the distances between observer, lens,
and source (which are determined from cosmological redshifts) will also affect the size of the Einstein ring, and the accuracy with which these distances are measured will impact the
uncertainty in the final mass estimate. In the gravity simulator,
these distances are relatively fixed, as massive lenses naturally
sink to the center. The effect of distances on θE can be explored
using the wine glass demonstration discussed at the end of this
article.
Gary D. White and Michael Walker, “The shape of ‘the Spandex’ and orbits upon its surface,” Am. J. Phys. 70, 48 (Jan. 2002).
Chad A. Middleton and Michael Langston, “Circular orbits on
a warped spandex fabric,” Am. J. Phys. 82, 287 (April 2014).
http://www.huffingtonpost.com/2013/12/04/physics-teacherdan-burns-gravity-visualized_n_4378960.html .
Melissa Hoffman and Meredith Woy, “Fabric of the Cosmos,”
2012 Science Outreach Catalyst Kit, http://www.spsnational.
org/programs/socks/2012.htm .
We used plywood to create the circular frame because we
GPVOEUIBU17$QJQFTXFSFUPPSJHJEUPCFOEJOUPBDJSDMFPGPVS
designated diameter. This is in contrast to experience discussed
in Ref. 8.
Here we used a localized central mass to investigate orbital rotation speeds. A more accurate model of dark matter would be
17.
18.
19.
20.
THE PHYSICS TEACHER ◆ Vol. 53, DECEMBER 2015
to use an extended object—perhaps a flexible sheet of lead—attached to the underside of the spandex, since the dark matter is
known to extend far beyond the visible matter of a galaxy (see
Fig. 1). However, our students do not appear to suffer misconceptions about dark matter detection because of this model
limitation.
In this respect, we started similarly to the activity “Introduction to Spandex as a Model for Spacetime,” described in the
2012 Science Outreach Catalyst Kit in Ref. 9.
In order to compare the Einstein radius of different lenses, it is
important to have a way to standardize the initial speed of the
photons.
The facilitator may need to remove the optional foam pipe
insulation around the edge of the simulator in order to allow
marbles to roll off the device and into the telescope bucket.
A similar activity, titled “Formation of the Solar System,” is described in Ref. 9.
1BVM)VXFBOE4DPUU'JFMEi.PEFSOHSBWJUBUJPOBMMFOTDPTNPMogy for introductory physics and astronomy students,” Phys.
Teach. 53, 266–270 (May 2015).
Maria Falbo-Kenkel and Joe Lohre, “Simple gravitational lens
demonstrations,” Phys. Teach. 34, 555–557 (Dec. 1996).
Don Lincoln, “Dark matter,” Phys. Teach. 51, 134–138 (March
2013).
http://universe.sonoma.edu/activities/dark_matter.html ,
http://www.nasa.gov/audience/forstudents/9-12/features/
what-is-dark-matter.html ,
http://home.web.cern.ch/about/physics/dark-matter , and
http://www.perimeterinstitute.ca/outreach/teachers/
multimedia-resources/mystery-dark-matter-video-game .
For elementary school students it could be a model of the solar
system, while for more advanced students it could be a visualization of Einstein’s curved spacetime.
Jared Stang is a Science Teaching and Learning Fellow in the
Department of Physics and Astronomy at the University of British
Columbia. Jared works in Physics Education Research and promotes and
supports faculty development in teaching. He has a background in theoretical high energy physics and is passionate about science outreach.
University of British Columbia, Department of Physics & Astronomy,
Vancouver, BC V6T 1Z1, Canada; [email protected]
Jes Ford is a Moore/Sloan Data Science and WRF Innovation in Data
Science Postdoctoral Fellow at the University of Washington. She completed her PhD at the University of British Columbia, studying dark matter
in the universe with gravitational lensing. Jes is excited about developing
engaging hands-on activities relevant to topics in cosmology and astrophysics.
[email protected]
Catherine Anderson is a clinical associate professor at the University
of British Columbia. She also currently runs Future Science Leaders, an
enrichment program at Science World that supports and challenges top
teens to excel in science and technology.
[email protected]
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